We will continue our focus on the number line and the relationships among the various types of numbers that make up the real number system. The following exercises will help you further understand the properties that hold true for each of the sets of numbers and the relationships among them.

As we saw in Session 1, the real number system is made up of many different sets. Some of these sets are quite large and contain other smaller sets. The integers, for example -- made up of the whole numbers and their negatives -- clearly contain the counting numbers (1, 2, 3, 4, ...). But which sets contain which other sets, and how do they all relate to one another? Let's explore.

Using the number line from Session 1 as a reference, draw a diagram that illustrates the relationships among the different sets of numbers that make up our number system -- the real numbers plus imaginary and complex numbers. Include all of the sets we discussed in Session 1:

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Counting numbers (1, 2, 3, 4, ...)

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Whole numbers (the counting numbers and 0)

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Integers (positive and negative whole numbers)

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Rational numbers (numbers that can be expressed as a ratio of two integers; when expressed in a decimal form, they will either terminate or repeat)

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Irrational numbers (numbers such as or e or square roots that can't be expressed as a ratio of two integers; they can be expressed as infinite, non-repeating decimals)

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Transcendental numbers (numbers that cannot be the solution to a polynomial equation; e.g., and e)

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Real numbers (all rational and irrational numbers; numbers that can be represented on a number line)

Pure imaginary numbers (multiples of i, a number such that when you square it, you get -1; e.g., 5i, 99i)

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Complex numbers (numbers created by the addition of imaginary and real number elements; e.g., 1 + 5i; 3/2 + 19i)

Use boxes or circles to represent each number set. Shapes that represent number sets should be placed within their larger set in the number system. Close Tip

Use boxes or circles to represent each number set. Shapes that represent number sets should be placed within their larger set in the number system. Compare and contrast your diagram with the diagram below, which shows one way to illustrate the relationships: